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Dependence of firing pattern on intrinsic ionic conductances: Sensitive and insensitive combinations
Neurocomputing 32}33 (2000) 141}146
Dependence of "ring pattern on intrinsic ionic conductances: Sensitive and insensitive combinations Mark S. Goldman *, Jorge Golowasch , Eve Marder , L.F. Abbott Volen Center for Complex Systems, Brandeis University, 415 South Street MS013 Waltham, MA 02454 USA Department of Physics, Harvard University, Cambridge, MA 02138 USA Accepted 13 January 2000
Abstract We construct maps of the intrinsic "ring state (silent, tonically "ring, or bursting) of a model neuron as a function of its maximal ionic conductances. The "ring properties vary signi"cantly only in response to changes in particular, sensitive combinations of conductances that may serve as targets for neuromodulation. Less sensitive combinations de"ne directions along which neuronal conductances may drift without triggering signi"cant changes in "ring activity. This suggests that neurons may have similar functional properties despite widely varying conductance densities. 2000 Elsevier Science B.V. All rights reserved. Keywords: Conductance-based model; Activity-dependent regulation; Neuromodulation; Stomatogastric ganglion
1. Introduction Electrophysiological measurements of ionic conductances show signi"cant cell-tocell variability [1,3]. Although traditionally attributed to artifacts of the recordings, we have shown that this variability may be a natural feature of how neurons are built [1], conferring onto them stability of activity while allowing for conductance variability resulting from turnover of channel proteins and their regulation. We have also shown that variability of the maximal conductances in a conductance-based model neuron does not necessarily imply variability in its activity patterns (Golowasch et al.,
* Correspondence address. Volen Center for Complex Systems, Brandeis University, 415 South Street MS013 Waltham, MA 02454 USA. Tel.: #1-781-736-3134; fax: #1-781-736-3142. E-mail address: [email protected] (M.S. Goldman). 0925-2312/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 0 0 ) 0 0 1 5 5 - 7
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M.S. Goldman et al. / Neurocomputing 32}33 (2000) 141}146
unpublished observations). Broad areas in a multidimensional parameter space de"ne characteristic activity patterns, with activity speci"ed by several conductances rather than by single conductances alone. In the present article we examine in more detail the contribution that each conductance makes to activity, both across activity pattern boundaries and, more speci"cally, within the regions of parameter space for each type of activity.
2. Methods A conductance-based model was built using standard Hodgkin}Huxley equations to describe each one of "ve voltage-dependent conductances (I , I , I , I , and , ) ! )! I ) and a leak current. Currents were based on measurements in the stomatogastric ganglion (STG) [5] and can be found in [3]. We "xed the maximum conductance corresponding to each of the currents at various di!erent values and identi"ed the resulting patterns of activity as either silent, tonic action potential "ring, or bursting. Cells with constant membrane potentials were classi"ed as silent. Cells with clusters of action potentials separated from each other by long silent intervals were classi"ed as bursters. Cells with regular action potential "ring were classi"ed as tonically "ring cells or as single-spike bursters, depending on their ability to cause transmission at a model synapse based on measurements of graded synaptic transmission within the STG [4].
Fig. 1. Variation in conductance densities in the crab PD neuron and in the model neuron. (A) Measurement of the three outward conductances in 13 di!erent PD neurons. Conductance densities vary over a factor of 2.5 for g , 2.2 for g , and 3.0 for g . (B) Firing state of the model neuron as a function of the ) )! three outward currents' maximal conductances. g Ca and g Na were varied randomly. The "ring state
cannot be determined by speci"cation of only the maximal conductances of the three outward currents. Intrinsically silent (black points), tonically "ring (gray points), and bursting (white points) activity patterns each "ll the entire region of parameter space over which these three conductances were varied.
M.S. Goldman et al. / Neurocomputing 32}33 (2000) 141}146
143
We measured the peak conductances of the three K> currents (I , I , and I ) ) )! expressed by isolated pyloric dilator (PD) neurons of the stomatogastric ganglion of the crab Cancer borealis in voltage clamp ([1], Fig. 1). Conductance densities were calculated by normalizing the peak conductances against the capacitance of the cell, which was measured by integrating the capacitive current during a brief voltage clamp test pulse [1].
3. Results Fig. 1 illustrates the complexity of the parameter space for the three outward conductances measured in real PD neurons (Fig. 1A), and for the same three conductances in the model neuron (Fig. 1B). Although PD cells possess a characteristic pattern of activity, the outward currents show a large degree of variability in their peak conductances (Fig. 1A). This suggests that these currents can vary within very wide margins without disrupting activity. The wide variability in individual conductances also suggests that measurement of a single conductance is not su$cient to determine the cell's activity state. We therefore tested whether measurement of single conductances, or even all three outward currents, was su$cient to characterize the intrinsic "ring state of the model neuron (Fig. 1B). We found that, while some trends existed, the measurement of the three outward conductances was insu$cient to specify the intrinsic "ring state (silent, tonic, or bursting). Each one of the three patterns of activity occupies the whole three-dimensional parameter space. Thus, no combination of outward currents determines uniquely a state of activity, and each activity pattern is highly permissive of variations in maximal conductance of the outward currents in the cell. In contrast, when another combination of conductances is examined (something that can only be done with a model cell), we "nd that the parameter space appears clearly structured (Fig. 2A). Regions with di!erent patterns of activity are reasonably well separated, though each state of activity is de"ned broadly by these parameters, and we observe slabs of parameter space occupied by each activity pattern. The maximal conductances shown, g Ca, g A, and g Kd, are found to be the most
important conductances in separating the tonic region of activity from the bursting region. That is, changes in a combination of these three conductances that point across the border between the tonic and bursting regions would be most e!ective in changing activity from tonic to bursting. In contrast, changes along directions that do not cross regional boundaries, or changes in conductances to which the neuron's activity pattern is not sensitive, could be quite large without producing a signi"cant modi"cation in activity. For example, changing g KCa did not strongly a!ect the
cell's activity, while replacing g Kd in the plot by g Na led to a slightly better
separation of the tonic points from the silent points (Goldman et al., unpublished observations). We next looked at details within each of the regions in parameter space (Fig. 2B}D). This allowed us to test whether the "ring properties within the "ring regions were sensitive to the same conductances that were important in distinguishing
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M.S. Goldman et al. / Neurocomputing 32}33 (2000) 141}146
Fig. 2. Dependence of the model neuron's activity patterns on its underlying maximal conductance densities. (A) Speci"cation of g A, g Ca, and g Kd leads to a strong separation of the "ring states into
particular regions of the parameter space. Shading is as in Fig. 1B. (B) Steady-state voltage levels for the intrinsically silent cells. (Black) !54.8 to !53.0 mV; (gray) !53.0 to !51.7 mV; (white) !51.7 to !45.5 mV. (C) Action potential "ring rate for tonically "ring cells. (Black) 0.35 to 1.10 Hz; (gray) 1.15 to 2.50 Hz; (white) 24.0 to 89.6 Hz. (D) Bursting rate for bursting cells. (Black) 0.2 to 1.04 Hz; (gray) 1.05 to 1.14 Hz; (white) 1.15 to 2.40 Hz.
between regions. We separated the silent cells into groups with di!erent levels of depolarization (Fig. 2B) and looked for sets of three conductances that revealed a clear separation of the di!erent levels. We found that the combination of g A, g Ca,
and g Kd best de"nes the steady-state level of depolarization of silent cells, while
other combinations are signi"cantly less adequate. The tonic region was next characterized by identifying the action potential "ring rate of the di!erent cells (Fig. 2C), and the best separation was obtained by a combination of g A, g Ca, and g KCa.
The KCa conductance had little e!ect in separating the low (black) and middle (gray)
M.S. Goldman et al. / Neurocomputing 32}33 (2000) 141}146
145
"ring rate cells. However, the high "ring rate neurons are all characterized by values of g KCa that are very close to zero. Inspection of the voltage traces of these cells
shows that their "ring is not a &classic' tonic "ring, but rather is characterized by a depolarized baseline membrane potential similar to the baseline membrane potential of the bursting neurons. These tonic cells are more properly labeled as bursters that did not have enough KCa current to repolarize them from the bursting plateau. Thus, it is not surprising that these points are located in a region of the conductance map that corresponds more closely to the location of the high burst-rate bursters (Fig. 2D, white points). Fig. 2D shows the best subset of three ionic currents that separates the bursting pattern of activity into di!erent bursting rates. We "nd that, in this case, the best separation was obtained by a combination of g A, g Ca, and g KCa.
Within the bursting region, g KCa is extremely important in determining both the
rate of bursting and the number of action potentials per burst (Goldman et al., unpublished observations). Thus, we "nd that a conductance that is relatively unimportant in determining some features of activity can be critically important in determining other features.
4. Discussion We have observed that the large variability in the maximal conductances of measured ionic currents is a natural feature of real cells ([2,4], and see results). Furthermore, we have shown in a conductance-based model that such variability in ionic currents can be consistent with and may indeed enable the production of stable patterns of activity. These results suggest that maintaining activity patterns does not require "ne tuning of the properties (i.e. parameters) of each individual ionic current to within narrow limits. Instead, they suggest that activity can be stable over a broad range of parameter values. In the case of our study, this stability can manifest itself in two ways. First, certain individual conductances were found to have relatively little e!ect on neural "ring properties. For example, changes in g KCa had very little
in#uence in determining whether a neuron was silent, tonic, or bursting. Such insensitivity to changes in the level of g was also observed in experimental )! manipulations of g in real neurons (Golowasch et al., unpublished observations). )! Second, certain changes in conductance combinations were found to have little in#uence due to apparently o!setting e!ects of the di!erent conductances. Thus, while adding g Ca and subtracting g A robustly changes the "ring state of the model
neuron, adding both g Ca and g A in combination may leave the neuron in the
same "ring state (Fig. 2A). We have shown that regulation of activity in a multidimensional conductancebased model occurs at two levels: One in which states of activity, such as silent, tonic "ring of action potentials, or bursting, are established in a general manner, and another in which the "ne details of a particular state of activity are de"ned. Certain conductances, such as g and g , are important in determining properties on ! both levels, with an increase of g Ca in combination with a decrease in g A
tending to push cells in a steady progression from silence towards bursting (Fig. 2A),
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M.S. Goldman et al. / Neurocomputing 32}33 (2000) 141}146
and from lower voltages (Fig. 2B) or "ring rates (Fig. 2C,D) to higher voltages or "ring rates, respectively. However, details of bursting are a!ected by quite di!erent conductances. Our results have interesting implications for the regulation of activity in biological systems. For instance, while a neuromodulator that evokes a very small ionic current may have a profound e!ect on the activity pattern generated by a neuron (Golowasch et al., unpublished observations), a di!erent neuromodulator that produces a larger change may have relatively little e!ect. It has been suggested previously that neural conductance values may lie near such borders to enhance the sensitivity of neurons to neuromodulators that move these conductances across such borders [2]. The broad and relatively well-de"ned borders that we have found between regions suggests that even neurons with conductances that are widely scattered along such borders can respond in the same, robust manner to the application of a neuromodulator. Thus, variability of conductances can be consistent with both stable "ring activity and with stable neuromodulatory action. Acknowledgements This work was supported by Grants MH 46742, NSF-IBN-9817194, and the W.M. Keck Foundation. References [1] J. Golowasch, L.F. Abbott, E. Marder, Activity-dependent regulation of potassium currents in an identi"ed neuron of the stomatogastric ganglion of the crab Cancer borealis, J. Neurosci. 19 (RC33) (1999) 1}5. [2] J. Guckenheimer, S. Gueron, R.M. Harris-Warrick, Mapping the dynamics of a bursting neuron. Phil. Trans. Roy. Soc. Lond. B 341 (1993) 345}359. [3] Z. Liu, J. Golowasch, E. Marder, L.F. Abbott, A model neuron with activity-dependent conductances regulated by multiple calcium sensors, J. Neurosci. 18 (1998) 2309}2320. [4] Y. Manor, F. Nadim, L.F. Abbott, E. Marder, Temporal dynamics of graded synaptic transmission in the lobster stomatogastric ganglion, J. Neurosci. 17 (1997) 5610}5621. [5] G. Turrigiano, G. LeMasson, E. Marder, Selective regulation of current densities underlies spontaneous changes in the activity of cultured neurons, J. Neurosci. 15 (1995) 3640}3652. Mark Goldman is a Ph.D. student in physics and computational neuroscience. His research focuses on the computational implications of activity-dependent neuronal processes. Jorge Golowasch is a senior research associate in neuroscience. His work focuses on the mechanisms of activity-dependent regulation of both ionic currents and activity patterns. Eve Marder is the Victor and Gwendolyn Beinfeld Professor of Neuroscience at Brandeis University. She studies the modulation of neurons and rhythmic networks. L.F. Abbott is the Nancy Lurie Marks Professor of Neuroscience and Director of the Volen Center at Brandeis University. His research covers a range of issues in computational neuroscience.
Dependence of "ring pattern on intrinsic ionic conductances: Sensitive and insensitive combinations Mark S. Goldman *, Jorge Golowasch , Eve Marder , L.F. Abbott Volen Center for Complex Systems, Brandeis University, 415 South Street MS013 Waltham, MA 02454 USA Department of Physics, Harvard University, Cambridge, MA 02138 USA Accepted 13 January 2000
Abstract We construct maps of the intrinsic "ring state (silent, tonically "ring, or bursting) of a model neuron as a function of its maximal ionic conductances. The "ring properties vary signi"cantly only in response to changes in particular, sensitive combinations of conductances that may serve as targets for neuromodulation. Less sensitive combinations de"ne directions along which neuronal conductances may drift without triggering signi"cant changes in "ring activity. This suggests that neurons may have similar functional properties despite widely varying conductance densities. 2000 Elsevier Science B.V. All rights reserved. Keywords: Conductance-based model; Activity-dependent regulation; Neuromodulation; Stomatogastric ganglion
1. Introduction Electrophysiological measurements of ionic conductances show signi"cant cell-tocell variability [1,3]. Although traditionally attributed to artifacts of the recordings, we have shown that this variability may be a natural feature of how neurons are built [1], conferring onto them stability of activity while allowing for conductance variability resulting from turnover of channel proteins and their regulation. We have also shown that variability of the maximal conductances in a conductance-based model neuron does not necessarily imply variability in its activity patterns (Golowasch et al.,
* Correspondence address. Volen Center for Complex Systems, Brandeis University, 415 South Street MS013 Waltham, MA 02454 USA. Tel.: #1-781-736-3134; fax: #1-781-736-3142. E-mail address: [email protected] (M.S. Goldman). 0925-2312/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 0 0 ) 0 0 1 5 5 - 7
142
M.S. Goldman et al. / Neurocomputing 32}33 (2000) 141}146
unpublished observations). Broad areas in a multidimensional parameter space de"ne characteristic activity patterns, with activity speci"ed by several conductances rather than by single conductances alone. In the present article we examine in more detail the contribution that each conductance makes to activity, both across activity pattern boundaries and, more speci"cally, within the regions of parameter space for each type of activity.
2. Methods A conductance-based model was built using standard Hodgkin}Huxley equations to describe each one of "ve voltage-dependent conductances (I , I , I , I , and , ) ! )! I ) and a leak current. Currents were based on measurements in the stomatogastric ganglion (STG) [5] and can be found in [3]. We "xed the maximum conductance corresponding to each of the currents at various di!erent values and identi"ed the resulting patterns of activity as either silent, tonic action potential "ring, or bursting. Cells with constant membrane potentials were classi"ed as silent. Cells with clusters of action potentials separated from each other by long silent intervals were classi"ed as bursters. Cells with regular action potential "ring were classi"ed as tonically "ring cells or as single-spike bursters, depending on their ability to cause transmission at a model synapse based on measurements of graded synaptic transmission within the STG [4].
Fig. 1. Variation in conductance densities in the crab PD neuron and in the model neuron. (A) Measurement of the three outward conductances in 13 di!erent PD neurons. Conductance densities vary over a factor of 2.5 for g , 2.2 for g , and 3.0 for g . (B) Firing state of the model neuron as a function of the ) )! three outward currents' maximal conductances. g Ca and g Na were varied randomly. The "ring state
cannot be determined by speci"cation of only the maximal conductances of the three outward currents. Intrinsically silent (black points), tonically "ring (gray points), and bursting (white points) activity patterns each "ll the entire region of parameter space over which these three conductances were varied.
M.S. Goldman et al. / Neurocomputing 32}33 (2000) 141}146
143
We measured the peak conductances of the three K> currents (I , I , and I ) ) )! expressed by isolated pyloric dilator (PD) neurons of the stomatogastric ganglion of the crab Cancer borealis in voltage clamp ([1], Fig. 1). Conductance densities were calculated by normalizing the peak conductances against the capacitance of the cell, which was measured by integrating the capacitive current during a brief voltage clamp test pulse [1].
3. Results Fig. 1 illustrates the complexity of the parameter space for the three outward conductances measured in real PD neurons (Fig. 1A), and for the same three conductances in the model neuron (Fig. 1B). Although PD cells possess a characteristic pattern of activity, the outward currents show a large degree of variability in their peak conductances (Fig. 1A). This suggests that these currents can vary within very wide margins without disrupting activity. The wide variability in individual conductances also suggests that measurement of a single conductance is not su$cient to determine the cell's activity state. We therefore tested whether measurement of single conductances, or even all three outward currents, was su$cient to characterize the intrinsic "ring state of the model neuron (Fig. 1B). We found that, while some trends existed, the measurement of the three outward conductances was insu$cient to specify the intrinsic "ring state (silent, tonic, or bursting). Each one of the three patterns of activity occupies the whole three-dimensional parameter space. Thus, no combination of outward currents determines uniquely a state of activity, and each activity pattern is highly permissive of variations in maximal conductance of the outward currents in the cell. In contrast, when another combination of conductances is examined (something that can only be done with a model cell), we "nd that the parameter space appears clearly structured (Fig. 2A). Regions with di!erent patterns of activity are reasonably well separated, though each state of activity is de"ned broadly by these parameters, and we observe slabs of parameter space occupied by each activity pattern. The maximal conductances shown, g Ca, g A, and g Kd, are found to be the most
important conductances in separating the tonic region of activity from the bursting region. That is, changes in a combination of these three conductances that point across the border between the tonic and bursting regions would be most e!ective in changing activity from tonic to bursting. In contrast, changes along directions that do not cross regional boundaries, or changes in conductances to which the neuron's activity pattern is not sensitive, could be quite large without producing a signi"cant modi"cation in activity. For example, changing g KCa did not strongly a!ect the
cell's activity, while replacing g Kd in the plot by g Na led to a slightly better
separation of the tonic points from the silent points (Goldman et al., unpublished observations). We next looked at details within each of the regions in parameter space (Fig. 2B}D). This allowed us to test whether the "ring properties within the "ring regions were sensitive to the same conductances that were important in distinguishing
144
M.S. Goldman et al. / Neurocomputing 32}33 (2000) 141}146
Fig. 2. Dependence of the model neuron's activity patterns on its underlying maximal conductance densities. (A) Speci"cation of g A, g Ca, and g Kd leads to a strong separation of the "ring states into
particular regions of the parameter space. Shading is as in Fig. 1B. (B) Steady-state voltage levels for the intrinsically silent cells. (Black) !54.8 to !53.0 mV; (gray) !53.0 to !51.7 mV; (white) !51.7 to !45.5 mV. (C) Action potential "ring rate for tonically "ring cells. (Black) 0.35 to 1.10 Hz; (gray) 1.15 to 2.50 Hz; (white) 24.0 to 89.6 Hz. (D) Bursting rate for bursting cells. (Black) 0.2 to 1.04 Hz; (gray) 1.05 to 1.14 Hz; (white) 1.15 to 2.40 Hz.
between regions. We separated the silent cells into groups with di!erent levels of depolarization (Fig. 2B) and looked for sets of three conductances that revealed a clear separation of the di!erent levels. We found that the combination of g A, g Ca,
and g Kd best de"nes the steady-state level of depolarization of silent cells, while
other combinations are signi"cantly less adequate. The tonic region was next characterized by identifying the action potential "ring rate of the di!erent cells (Fig. 2C), and the best separation was obtained by a combination of g A, g Ca, and g KCa.
The KCa conductance had little e!ect in separating the low (black) and middle (gray)
M.S. Goldman et al. / Neurocomputing 32}33 (2000) 141}146
145
"ring rate cells. However, the high "ring rate neurons are all characterized by values of g KCa that are very close to zero. Inspection of the voltage traces of these cells
shows that their "ring is not a &classic' tonic "ring, but rather is characterized by a depolarized baseline membrane potential similar to the baseline membrane potential of the bursting neurons. These tonic cells are more properly labeled as bursters that did not have enough KCa current to repolarize them from the bursting plateau. Thus, it is not surprising that these points are located in a region of the conductance map that corresponds more closely to the location of the high burst-rate bursters (Fig. 2D, white points). Fig. 2D shows the best subset of three ionic currents that separates the bursting pattern of activity into di!erent bursting rates. We "nd that, in this case, the best separation was obtained by a combination of g A, g Ca, and g KCa.
Within the bursting region, g KCa is extremely important in determining both the
rate of bursting and the number of action potentials per burst (Goldman et al., unpublished observations). Thus, we "nd that a conductance that is relatively unimportant in determining some features of activity can be critically important in determining other features.
4. Discussion We have observed that the large variability in the maximal conductances of measured ionic currents is a natural feature of real cells ([2,4], and see results). Furthermore, we have shown in a conductance-based model that such variability in ionic currents can be consistent with and may indeed enable the production of stable patterns of activity. These results suggest that maintaining activity patterns does not require "ne tuning of the properties (i.e. parameters) of each individual ionic current to within narrow limits. Instead, they suggest that activity can be stable over a broad range of parameter values. In the case of our study, this stability can manifest itself in two ways. First, certain individual conductances were found to have relatively little e!ect on neural "ring properties. For example, changes in g KCa had very little
in#uence in determining whether a neuron was silent, tonic, or bursting. Such insensitivity to changes in the level of g was also observed in experimental )! manipulations of g in real neurons (Golowasch et al., unpublished observations). )! Second, certain changes in conductance combinations were found to have little in#uence due to apparently o!setting e!ects of the di!erent conductances. Thus, while adding g Ca and subtracting g A robustly changes the "ring state of the model
neuron, adding both g Ca and g A in combination may leave the neuron in the
same "ring state (Fig. 2A). We have shown that regulation of activity in a multidimensional conductancebased model occurs at two levels: One in which states of activity, such as silent, tonic "ring of action potentials, or bursting, are established in a general manner, and another in which the "ne details of a particular state of activity are de"ned. Certain conductances, such as g and g , are important in determining properties on ! both levels, with an increase of g Ca in combination with a decrease in g A
tending to push cells in a steady progression from silence towards bursting (Fig. 2A),
146
M.S. Goldman et al. / Neurocomputing 32}33 (2000) 141}146
and from lower voltages (Fig. 2B) or "ring rates (Fig. 2C,D) to higher voltages or "ring rates, respectively. However, details of bursting are a!ected by quite di!erent conductances. Our results have interesting implications for the regulation of activity in biological systems. For instance, while a neuromodulator that evokes a very small ionic current may have a profound e!ect on the activity pattern generated by a neuron (Golowasch et al., unpublished observations), a di!erent neuromodulator that produces a larger change may have relatively little e!ect. It has been suggested previously that neural conductance values may lie near such borders to enhance the sensitivity of neurons to neuromodulators that move these conductances across such borders [2]. The broad and relatively well-de"ned borders that we have found between regions suggests that even neurons with conductances that are widely scattered along such borders can respond in the same, robust manner to the application of a neuromodulator. Thus, variability of conductances can be consistent with both stable "ring activity and with stable neuromodulatory action. Acknowledgements This work was supported by Grants MH 46742, NSF-IBN-9817194, and the W.M. Keck Foundation. References [1] J. Golowasch, L.F. Abbott, E. Marder, Activity-dependent regulation of potassium currents in an identi"ed neuron of the stomatogastric ganglion of the crab Cancer borealis, J. Neurosci. 19 (RC33) (1999) 1}5. [2] J. Guckenheimer, S. Gueron, R.M. Harris-Warrick, Mapping the dynamics of a bursting neuron. Phil. Trans. Roy. Soc. Lond. B 341 (1993) 345}359. [3] Z. Liu, J. Golowasch, E. Marder, L.F. Abbott, A model neuron with activity-dependent conductances regulated by multiple calcium sensors, J. Neurosci. 18 (1998) 2309}2320. [4] Y. Manor, F. Nadim, L.F. Abbott, E. Marder, Temporal dynamics of graded synaptic transmission in the lobster stomatogastric ganglion, J. Neurosci. 17 (1997) 5610}5621. [5] G. Turrigiano, G. LeMasson, E. Marder, Selective regulation of current densities underlies spontaneous changes in the activity of cultured neurons, J. Neurosci. 15 (1995) 3640}3652. Mark Goldman is a Ph.D. student in physics and computational neuroscience. His research focuses on the computational implications of activity-dependent neuronal processes. Jorge Golowasch is a senior research associate in neuroscience. His work focuses on the mechanisms of activity-dependent regulation of both ionic currents and activity patterns. Eve Marder is the Victor and Gwendolyn Beinfeld Professor of Neuroscience at Brandeis University. She studies the modulation of neurons and rhythmic networks. L.F. Abbott is the Nancy Lurie Marks Professor of Neuroscience and Director of the Volen Center at Brandeis University. His research covers a range of issues in computational neuroscience.